A new PDE approach for pricing arithmetic average Asian options
|
|
- Beatrix Heath
- 6 years ago
- Views:
Transcription
1 A new PDE approach for pricing arithmetic average Asian options Jan Večeř Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA May 15, 21 Abstract. In this paper, arithmetic average Asian options are studied. It is observed that the Asian option is a special case of the option on a traded account. The price of the Asian option is characterized by a simple one-dimensional partial differential equation which could be applied to both continuous and discrete average Asian option. The article also provides numerical implementation of the pricing equation. The implementation is fast and accurate even for low volatility and/or short maturity cases. Key words: Asian options, Options on a traded account, Brownian motion, fixed strike, floating strike. 1 Introduction Asian options are securities with payoff which depends on the average of the underlying stock price over certain time interval. Since no general analytical solution for the price of the Asian option is known, a variety of techniques have been developed to analyze arithmetic average Asian options. A number of approximations that produce closed form expressions have appeared, see Turnbull and Wakeman [18], Vorst [19], Levy [13], Levy and Turnbull [14]. Geman and Yor [8] computed the Laplace transform of the Asian option price, but numerical inversion remains problematic for low volatility and/or short maturity cases (see Geman and Eydeland [6] or Fu, Madan and Wang [5]). Monte Carlo simulation works well, but it can be computationally expensive without the enhancement of variance reduction techniques and one must account for the inherent discretization bias resulting from the approximation of continuous time processes through discrete sampling (see Broadie and Glasserman [3], Broadie, Glasserman and Kou [4] and Kemma and Vorst [12]). This work was supported by the National Science Foundation under grant DMS I would like to thank Fredrik Åkesson, Julien Hugonnier, Steven Shreve, Dennis Wong and Mingxin Xu for helpful comments and suggestions on this paper. 1
2 In general, the price of an Asian option can be found by solving a PDE in two space dimensions (see Ingersoll [1]), which is prone to oscillatory solutions. Ingersoll [1] also observed that the two-dimensional PDE for a floating strike Asian option can be reduced to a one-dimensional PDE. Rogers and Shi [17] have formulated a one-dimensional PDE that can model both floating and fixed strike Asian options. They reduced the dimension of the problem by dividing K S t (K is the strike, S t is the average stock price over [, t]) by the stock price S t. However this one-dimensional PDE is difficult to solve numerically since the diffusion term is very small for values of interest on the finite difference grid. The dirac delta function also appears as a coefficient of the PDE in the case of the floating strike option. Zvan, Forsyth and Vetzal [21] were able to improve the numerical accuracy of this method by using computational fluid dynamics techniques. Andreasen [2] applied Rogers and Shi s reduction to discretely sampled Asian option. More recently, Lipton [15] noticed similarity of pricing equations for the passport and the Asian option, again using Rogers and Shi s reduction. In this article, an alternative one-dimensional PDE is derived by a similar space reduction. It is noted that the arithmetic average Asian option (both floating and fixed strike) is a special case of an option on a traded account. See Shreve and Večeř [16] and [2] for a detailed discussion about options on a traded account. Options on a traded account generalize the concept of many options (passport, European, American, vacation) and the same pricing techniques could be applied to price the Asian option. The resulting one-dimensional PDE for the price of the Asian option is simple enough to be easily implemented to give very fast and accurate results. Section 2 of the article briefly describes options on a traded account. It is shown in section 3 that the Asian option is a special case of the option on a traded account. The one-dimensional PDE for the price of the Asian option is given. Section 4 describes the numerical implementation and compares results with results of other methods. Section 5 concludes the paper. 2 Options on a traded account An option on a traded account is a contract which allows the holder of the option to switch during the life of the option among various positions in an underlying asset (stock). The holder accumulates gains and losses resulting from this trading, and at the expiration of the option he gets the call option payoff with strike on his final account value, i.e., he keeps any gain from trading and is forgiven any loss. Suppose that the stock evolves under the risk neutral measure according to the equation ds t = S t (rdt + σdw t ), (2.1) where r is the interest rate and σ is the volatility of the stock. Denote the option holder s trading strategy by q t, the number of shares held at time t. The strategy q t is subject to the contractual constraint q t [α t, β t ], where α t β t. It turns out that the holder of the option should never take an intermediate position, i.e, 2
3 at any time he should hold either α t shares of stock or β t shares. In the case of Asian options, α t = β t, so option holder s trading strategy is a priori given to him. In our model the value of the option holder s account corresponding to the strategy q t satisfies dx q t = q t ds t + µ(x q t q t S t )dt (2.2) X q = X. This represents a trading strategy in the money market and the underlying asset, where X is the initial wealth and µ is the interest rate corresponding to reinvesting the cash position X q t q t S t (possibly different from the risk-neutral interest rate r). The trading strategy is self-financing when µ = r. The holder of the option will receive at time T the payoff [X q T ]+. The objective of the seller of the option, who makes this payment, is to be prepared to hedge against all possible strategies of the holder of the option. Therefore the price of this contract at time t should be the maximum over all possible strategies q u of the discounted expected value under the risk-neutral probability P of the payoff of the option, i.e., V [α,β] (t, S t, X t ) = max q e r(t t) E[[X q T ]+ F t ], t [, T ]. (2.3) u [α,β] Computation of the expression in (2.3) is a problem of stochastic optimal control, and the function V [α,β] (t, s, x) is characterized by the corresponding Hamilton Jacobi Bellman (HJB) equation rv + V t + rsv s + max q [α,β] [(µx + q(r µ))v x with the boundary condition σ2 s 2 (V ss + 2qV sx + q 2 V xx )] = (2.4) V (T, s, x) = x +. (2.5) The maximum in (2.4) is attained by the optimal strategy q opt t. The case α t = β t = 1 reduces to the European call, the case α t = β t = 1 reduces to the European put. The American call and put give the holder of the option the right to switch at most once during the life of the option to zero position (i.e., exercise the option), but it does not pay interest on the traded account while the holder has a position in the stock market. These can be modelled by setting µ = in (2.2) and allowing only one switch in q t, either from 1 to (American call) or from 1 to (American put). The passport option has contractual conditions α t = 1, β t = 1, the so-called vacation call has α t =, β t = 1 and the so-called vacation put has α t = 1, β t =. By the change of variable Z q t = Xq t S t, (2.6) 3
4 we can reduce the dimensionality of the problem (2.3), as we show below. The same change of variable was used in Hyer, Lipton-Lifschitz and Pugachevsky [9] and in Andersen, Andreasen and Brotherton-Ratcliffe [1] to price passport options and in Shreve and Večeř [16] to price options on a traded account. Applying Itô s formula to the process Z q t, we get dz q t = (q t Z q t ) (r µ σ 2 )dt + (q t Z q t ) σdw t. (2.7) We next define a new probability measure P by P(A) = A D T dp, A F, where D T = e rt ST S = exp ( σw T 1 2 σ2 T ). (2.8) Under P, W t = σt+w t is a Brownian motion, according to Girsanov s theorem. Notice that [ e rt E[X q T ]+ = e rt X q ] + [ T Ẽ D T = S Ẽ X q ] + T S T = S Ẽ [Zq T ]+ (2.9) and dz q t = (q t Z q t ) (r µ)dt + (q t Z q t ) σd W t. (2.1) The corresponding reduced HJB equation becomes ( u t + max (r µ)(q z)uz + 1 q [α,β] 2 (q z)2 σ 2 ) u zz = (2.11) with the boundary condition The relationship between V and u is u(t, z) = z +. (2.12) V (, S, X ) = S u ( ), X S. (2.13) Closed form solutions and optimal strategies are provided in Shreve and Večeř [16] for the prices of the option on a traded account for any general constraints of the type α t α and β t β when µ = r. 3 Asian option as an option on a traded account Options on a traded account also represent Asian options. Notice that d(ts t ) = tds t + S t dt, or equivalently, T S T = tds t + S t dt. (3.1) After dividing by the maturity time T and rearranging the terms we get 1 T S t dt = ( 1 t T ) dst + S. (3.2) 4
5 In the terminology of the option on a traded account, the Asian fixed strike call payoff ( S T K) + is achieved by taking q t = 1 t T and X = S K and where the traded account evolves according to the equation dx t = ( 1 t T ) dst, (3.3) i.e., when µ = so no interest is added or charged to the traded account. We have then X T = (1 t T )ds t + S K = S T K. (3.4) Thus the average of the stock price could be achieved by a selling off one share of stock at the constant rate 1 T shares per unit time. Similarly, the Asian fixed strike put payoff (K S T ) + is achieved by taking q t = t T 1 and X = K S. For the Asian floating strike call with payoff (KS T S T ) + we take simply q t = t T 1 + K and X = S (K 1), for the Asian floating strike put with payoff ( S T KS T ) + we take q t = t T + 1 K and X = S (1 K). The discrete average Asian option payoff could be achieved by taking a step function approximation of the stock position q t of its continous average option counterpart. Let us take for example the case of the Asian fixed strike call when q t = 1 t T and X = S K. A step function approximation of 1 t T is [ ] q t = 1 1 n n t T, (3.5) where [ ] denotes the integer part function. If we look directly at the Asian option traded account equation we get for the stock position q t given by (3.5) X T = 1 n dx t = q t ds t, (3.6) n k=1 S ( k n ) T S + X. (3.7) Thus we get the discrete average Asian fixed strike call payoff ( 1 n n k=1 S ( k n ) T K ) + (3.8) [ ] by taking X = S K and q t = 1 1 n n t T. We get analogous results for other Asian option types. Since we showed that Asian options are options on a traded account, we can apply the same pricing techniques to determine the price of Asian options. In particular, we can use the HJB equation (2.11), which becomes for the case of Asian options just a simple PDE u t + r(q t z)u z (q t z) 2 σ 2 u zz = (3.9) 5
6 Asian option type Payoff Stock position q t Initial wealth X Fixed strike call ( S T K) + 1 t T S K Fixed strike put (K S T ) + t T 1 K S Floating strike call (KS T S T ) + t T 1 + K S (K 1) Floating strike put ( S T KS T ) + t T + 1 K S (1 K) Table 1: Asian options as options on a traded account with the boundary condition u(t, z) = z +. (3.1) The price of the Asian option is then given in terms of u by (2.13). The relationship between different kinds of Asian options and options on a traded account is summarized in Table 1. 4 Numerical examples Since there is very little hope that the partial differential equation (3.9) with the boundary condition (3.1) admits a closed form solution, one must compute the price of the Asian option numerically. Equation (3.9) is on the other hand very easy to implement and since it is an equation of the Black-Scholes type, it is also very stable and fast to compute. Numerical implementation of this PDE gives answers within very tight analytical bounds even for low volatility or short maturity contracts. The numerical implementation of the Asian option PDE (3.9) is similar to the numerical implementation for the passport option as described in Andersen, Andreasen and Brotherton-Ratcliffe [1] because of the above mentioned similarity in the pricing equation for both options. The Asian option pricing is even simpler compared to the passport option pricing. The reason is that the position in the stock q t is deterministically given for the Asian option, while the optimal position q t must be computed for the case of the passport option. Results obtained in Andersen et al. [1] for the case of the passport option show that the numerical implementation gave almost indistinguishable results from the analytical solution (less than.1% off) within less than a second of CPU time on 166 MHz Pentium. Let us consider a finite difference discretization of PDE (3.9) with a uniform mesh z i = z + i dz, t j = j dt for i M, j N, and t N = T, where z and z M represent and. Reasonable choices are z = 1 and z M = 1. One point represents the Asian option with strike equal to zero, the other represents the Asian option with strike equal to double of the stock price. Using the short notation u i,j = u(t j, z i ) and 6
7 q j = q(t j ), a mixed implicit/explicit finite discretization scheme for (3.9) is given by θ[σ 2 (q j z i ) 2 dz r(q j z i )]u i 1,j 2[θσ 2 (q j z i ) 2 + ν]u i,j + θ[σ 2 (q j z i ) 2 + dz r(q j z i )]u i+1,j = (1 θ)[σ 2 (q j z i ) 2 dz r(q j z i )]u i 1,j+1 + 2[(1 θ)σ 2 (q j z i ) 2 ν]u i,j+1 where θ 1 and ν = dz2 equations is (1 θ)[σ 2 (q j z i ) 2 + dz r(q j z i )]u i+1,j+1, (4.1) dt. The boundary condition for this system of u i,n = z + i. (4.2) Solving for u i,j is done in the usual way by solving the corresponding tridiagonal system of equations in (4.1). For the boundary conditions at z and z M we can take u,j = and u M,j = 2u M 1,j u M 2,j (linear interpolation). The parameter θ in (4.1) determines at what time point the partial derivatives with respect to z are evaluated. If θ =, the z derivatives are evaluated at t j+1 and the scheme is known as the explicit finite difference method or as a trinomial tree. If θ = 1, the z derivatives are evaluated at t j and the scheme becomes fully implicit finite difference method. The average of these two methods, i.e., when θ = 1 2, is known as a Crank-Nicolson method. Crank-Nicolson method is usually preferred, because it has the highest convergence order in dt. This method was used to get numerical results in this article. Table 2 compares results of the above described method with results of Rogers and Shi [17], Zvan, Forsyth and Vetzal [21] and with Monte Carlo methods. The comparison for the fixed strike Asian call when r =.15, S = 1 and T = 1, which they considered as the most difficult case is reported. Zvan et al. improved the accuracy of the method of Rogers and Shi by using a nonuniform spatial grid and techniques of computational fluid dynamics. To be consistent with the result of Zvan et al., same number of points of space and time grid (2 space points, 4 time points) are used in (4.1). The Monte Carlo method used here as a comparison uses techniques from Glasserman, Heidelberger and Shahabuddin [7] together with Sobol numbers and geometric Asian call option as control variate, which both reduces the variance and the bias from the discretization (see Fu, Madan and Wang [5]). The lower and upper analytical bounds mentioned here are according to Rogers and Shi. As seen from the table, the accuracy of the method suggested in this article is very good; it always gives prices within analytical bounds. It is stable for low volatilities and short maturities contrary to numerical inversion of the Laplace transform of the Asian option price or to other PDE methods for the Asian option. This implementation, done in MATLAB, gave accurate results in a few seconds. 7
8 σ K Večeř Zvan et al. Monte Carlo Lower Upper Table 2: Comparison of results of different methods for fixed strike Asian call when r =.15, S = 1 and T = 1. The upper and lower bounds were obtained from Rogers and Shi [17]. 5 Conclusion The pricing method for Asian options suggested in this article connects pricing of Asian options and options on a traded account. Options on a traded account (passport, European, American, vacation, Asian) satisfy the same type of onedimensional PDE. The method suggested here has a simple form, is easy to implement, has stable performance for all volatilities, is fast and accurate, and is applicable for both continuous and discrete average Asian options. References [1] Andersen, L., Andreasen, J., Brotherton-Ratcliffe, R., The passport option, The Journal of Computational Finance Vol. 1, No. 3, Spring 1998, [2] Andreasen, J. The pricing of discretely sampled Asian and lookback options: a change of numeraire approach, The Journal of Computational Finance Vol. 2, No. 1, Fall 1998, 5 3. [3] Broadie, M., Glasserman, P., Estimating security price derivatives using simulation, Management Science, 42, 1996, [4] Broadie, M., Glasserman, P., Kou, S. Connecting discrete and continuous path-dependent options, Finance and Stochastics, 3, 1999,
9 [5] Fu, M., Madan, D., Wang, T., Pricing continuous Asian options: a comparison of Monte Carlo and Laplace transform inversion methods, The Journal of Computational Finance, Vol. 2, No. 2, Winter 1998/99. [6] Geman, H., Eydeland, A., Domino effect, Risk, April 1995, [7] Glasserman, P., Heidelberger, P., Shahabuddin, Asymptotically optimal importance sampling and stratification for pricing path-dependent options, Mathemacical Finance, April 1999, [8] Geman, H., Yor, M., Bessel processes, Asian option, and perpetuities, Mathematical Finance, 3, 1993, [9] Hyer, T., Lipton-Lifschitz, A., Pugachevsky, D., Passport to success, Risk, Vol. 1, No. 9, September 1997, [1] Ingersoll, J., Theory of Financial Decision Making, Oxford, [11] Karatzas, I., Shreve, S., Brownian Motion and Stochastic Calculus, Springer Verlag, Second Edition, [12] Kemma, A., Vorst, A., A pricing method for options based on average asset values, Journal of Banking and Finance, 14, 199, [13] Levy, E., Pricing European average rate currency options, Journal of International Money and Finance, 11, 1992, [14] Levy, E., Turnbull, S., Average Inteligence, Risk, February 1992, [15] Lipton, A., Similarities via self-similarities, Risk, September 1999, [16] Shreve, S., Večeř, J., Options on a traded account: Vacation calls, vacation puts and passport options, Finance and Stochastics, 2. [17] Rogers, L., Shi, Z., The value of an Asian option, Journal of Applied Probability, 32, 1995, [18] Turnbull, S., Wakeman, L., A quick algorith for pricing European average options,, Journal of Financial and Quantitative Analysis, 26, 1991, [19] Vorst, T., Prices and hedge ratios of average exchange rate options, International Review of Financial Analysis, 1, 1992, [2] Večeř, J., Shreve, S., Upgrading your passport, Risk, July 2, [21] Zvan, R., Forsyth, P., Vetzal, K., Robust numerical methods for PDE models of Asian options, The Journal of Computational Finance, Vol. 1, No. 2, Winter 1997/98,
2.1 Mathematical Basis: Risk-Neutral Pricing
Chapter Monte-Carlo Simulation.1 Mathematical Basis: Risk-Neutral Pricing Suppose that F T is the payoff at T for a European-type derivative f. Then the price at times t before T is given by f t = e r(t
More informationPDE Methods for the Maximum Drawdown
PDE Methods for the Maximum Drawdown Libor Pospisil, Jan Vecer Columbia University, Department of Statistics, New York, NY 127, USA April 1, 28 Abstract Maximum drawdown is a risk measure that plays an
More informationMATH6911: Numerical Methods in Finance. Final exam Time: 2:00pm - 5:00pm, April 11, Student Name (print): Student Signature: Student ID:
MATH6911 Page 1 of 16 Winter 2007 MATH6911: Numerical Methods in Finance Final exam Time: 2:00pm - 5:00pm, April 11, 2007 Student Name (print): Student Signature: Student ID: Question Full Mark Mark 1
More informationCRANK-NICOLSON SCHEME FOR ASIAN OPTION
CRANK-NICOLSON SCHEME FOR ASIAN OPTION By LEE TSE YUENG A thesis submitted to the Department of Mathematical and Actuarial Sciences, Faculty of Engineering and Science, Universiti Tunku Abdul Rahman, in
More informationAN IMPROVED BINOMIAL METHOD FOR PRICING ASIAN OPTIONS
Commun. Korean Math. Soc. 28 (2013), No. 2, pp. 397 406 http://dx.doi.org/10.4134/ckms.2013.28.2.397 AN IMPROVED BINOMIAL METHOD FOR PRICING ASIAN OPTIONS Kyoung-Sook Moon and Hongjoong Kim Abstract. We
More informationAMH4 - ADVANCED OPTION PRICING. Contents
AMH4 - ADVANCED OPTION PRICING ANDREW TULLOCH Contents 1. Theory of Option Pricing 2 2. Black-Scholes PDE Method 4 3. Martingale method 4 4. Monte Carlo methods 5 4.1. Method of antithetic variances 5
More informationMonte Carlo Simulations
Monte Carlo Simulations Lecture 1 December 7, 2014 Outline Monte Carlo Methods Monte Carlo methods simulate the random behavior underlying the financial models Remember: When pricing you must simulate
More informationEFFICIENT MONTE CARLO ALGORITHM FOR PRICING BARRIER OPTIONS
Commun. Korean Math. Soc. 23 (2008), No. 2, pp. 285 294 EFFICIENT MONTE CARLO ALGORITHM FOR PRICING BARRIER OPTIONS Kyoung-Sook Moon Reprinted from the Communications of the Korean Mathematical Society
More informationAn Adjusted Trinomial Lattice for Pricing Arithmetic Average Based Asian Option
American Journal of Applied Mathematics 2018; 6(2): 28-33 http://www.sciencepublishinggroup.com/j/ajam doi: 10.11648/j.ajam.20180602.11 ISSN: 2330-0043 (Print); ISSN: 2330-006X (Online) An Adjusted Trinomial
More informationMATH4143: Scientific Computations for Finance Applications Final exam Time: 9:00 am - 12:00 noon, April 18, Student Name (print):
MATH4143 Page 1 of 17 Winter 2007 MATH4143: Scientific Computations for Finance Applications Final exam Time: 9:00 am - 12:00 noon, April 18, 2007 Student Name (print): Student Signature: Student ID: Question
More information1.1 Basic Financial Derivatives: Forward Contracts and Options
Chapter 1 Preliminaries 1.1 Basic Financial Derivatives: Forward Contracts and Options A derivative is a financial instrument whose value depends on the values of other, more basic underlying variables
More information13.3 A Stochastic Production Planning Model
13.3. A Stochastic Production Planning Model 347 From (13.9), we can formally write (dx t ) = f (dt) + G (dz t ) + fgdz t dt, (13.3) dx t dt = f(dt) + Gdz t dt. (13.33) The exact meaning of these expressions
More informationPricing Asian Options in a Semimartingale Model
Pricing Asian Options in a Semimartingale Model Jan Večeř Columbia University, Department of Statistics, New York, NY 127, USA Kyoto University, Kyoto Institute of Economic Research, Financial Engineering
More informationCS 774 Project: Fall 2009 Version: November 27, 2009
CS 774 Project: Fall 2009 Version: November 27, 2009 Instructors: Peter Forsyth, paforsyt@uwaterloo.ca Office Hours: Tues: 4:00-5:00; Thurs: 11:00-12:00 Lectures:MWF 3:30-4:20 MC2036 Office: DC3631 CS
More informationHedging with Life and General Insurance Products
Hedging with Life and General Insurance Products June 2016 2 Hedging with Life and General Insurance Products Jungmin Choi Department of Mathematics East Carolina University Abstract In this study, a hybrid
More informationMath 416/516: Stochastic Simulation
Math 416/516: Stochastic Simulation Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 13 Haijun Li Math 416/516: Stochastic Simulation Week 13 1 / 28 Outline 1 Simulation
More informationKing s College London
King s College London University Of London This paper is part of an examination of the College counting towards the award of a degree. Examinations are governed by the College Regulations under the authority
More informationNumerical Methods in Option Pricing (Part III)
Numerical Methods in Option Pricing (Part III) E. Explicit Finite Differences. Use of the Forward, Central, and Symmetric Central a. In order to obtain an explicit solution for the price of the derivative,
More informationA Moment Matching Approach To The Valuation Of A Volume Weighted Average Price Option
A Moment Matching Approach To The Valuation Of A Volume Weighted Average Price Option Antony Stace Department of Mathematics and MASCOS University of Queensland 15th October 2004 AUSTRALIAN RESEARCH COUNCIL
More informationMulti-Asset Options. A Numerical Study VILHELM NIKLASSON FRIDA TIVEDAL. Master s thesis in Engineering Mathematics and Computational Science
Multi-Asset Options A Numerical Study Master s thesis in Engineering Mathematics and Computational Science VILHELM NIKLASSON FRIDA TIVEDAL Department of Mathematical Sciences Chalmers University of Technology
More informationKing s College London
King s College London University Of London This paper is part of an examination of the College counting towards the award of a degree. Examinations are governed by the College Regulations under the authority
More informationBluff Your Way Through Black-Scholes
Bluff our Way Through Black-Scholes Saurav Sen December 000 Contents What is Black-Scholes?.............................. 1 The Classical Black-Scholes Model....................... 1 Some Useful Background
More informationA Continuity Correction under Jump-Diffusion Models with Applications in Finance
A Continuity Correction under Jump-Diffusion Models with Applications in Finance Cheng-Der Fuh 1, Sheng-Feng Luo 2 and Ju-Fang Yen 3 1 Institute of Statistical Science, Academia Sinica, and Graduate Institute
More informationImplementing Models in Quantitative Finance: Methods and Cases
Gianluca Fusai Andrea Roncoroni Implementing Models in Quantitative Finance: Methods and Cases vl Springer Contents Introduction xv Parti Methods 1 Static Monte Carlo 3 1.1 Motivation and Issues 3 1.1.1
More informationBasic Concepts in Mathematical Finance
Chapter 1 Basic Concepts in Mathematical Finance In this chapter, we give an overview of basic concepts in mathematical finance theory, and then explain those concepts in very simple cases, namely in the
More informationRichardson Extrapolation Techniques for the Pricing of American-style Options
Richardson Extrapolation Techniques for the Pricing of American-style Options June 1, 2005 Abstract Richardson Extrapolation Techniques for the Pricing of American-style Options In this paper we re-examine
More informationThe Use of Importance Sampling to Speed Up Stochastic Volatility Simulations
The Use of Importance Sampling to Speed Up Stochastic Volatility Simulations Stan Stilger June 6, 1 Fouque and Tullie use importance sampling for variance reduction in stochastic volatility simulations.
More informationThe Pennsylvania State University. The Graduate School. Department of Industrial Engineering AMERICAN-ASIAN OPTION PRICING BASED ON MONTE CARLO
The Pennsylvania State University The Graduate School Department of Industrial Engineering AMERICAN-ASIAN OPTION PRICING BASED ON MONTE CARLO SIMULATION METHOD A Thesis in Industrial Engineering and Operations
More informationLecture 4. Finite difference and finite element methods
Finite difference and finite element methods Lecture 4 Outline Black-Scholes equation From expectation to PDE Goal: compute the value of European option with payoff g which is the conditional expectation
More informationDefinition Pricing Risk management Second generation barrier options. Barrier Options. Arfima Financial Solutions
Arfima Financial Solutions Contents Definition 1 Definition 2 3 4 Contenido Definition 1 Definition 2 3 4 Definition Definition: A barrier option is an option on the underlying asset that is activated
More informationTHE OPTIMAL ASSET ALLOCATION PROBLEMFOR AN INVESTOR THROUGH UTILITY MAXIMIZATION
THE OPTIMAL ASSET ALLOCATION PROBLEMFOR AN INVESTOR THROUGH UTILITY MAXIMIZATION SILAS A. IHEDIOHA 1, BRIGHT O. OSU 2 1 Department of Mathematics, Plateau State University, Bokkos, P. M. B. 2012, Jos,
More informationContents Critique 26. portfolio optimization 32
Contents Preface vii 1 Financial problems and numerical methods 3 1.1 MATLAB environment 4 1.1.1 Why MATLAB? 5 1.2 Fixed-income securities: analysis and portfolio immunization 6 1.2.1 Basic valuation of
More informationThe Performance of Analytical Approximations for the Computation of Asian Quanto-Basket Option Prices
1 The Performance of Analytical Approximations for the Computation of Asian Quanto-Basket Option Prices Jean-Yves Datey Comission Scolaire de Montréal, Canada Geneviève Gauthier HEC Montréal, Canada Jean-Guy
More informationAn Efficient Numerical Scheme for Simulation of Mean-reverting Square-root Diffusions
Journal of Numerical Mathematics and Stochastics,1 (1) : 45-55, 2009 http://www.jnmas.org/jnmas1-5.pdf JNM@S Euclidean Press, LLC Online: ISSN 2151-2302 An Efficient Numerical Scheme for Simulation of
More informationRisk Minimization Control for Beating the Market Strategies
Risk Minimization Control for Beating the Market Strategies Jan Večeř, Columbia University, Department of Statistics, Mingxin Xu, Carnegie Mellon University, Department of Mathematical Sciences, Olympia
More informationMonte Carlo Methods in Financial Engineering
Paul Glassennan Monte Carlo Methods in Financial Engineering With 99 Figures
More informationLecture Note 8 of Bus 41202, Spring 2017: Stochastic Diffusion Equation & Option Pricing
Lecture Note 8 of Bus 41202, Spring 2017: Stochastic Diffusion Equation & Option Pricing We shall go over this note quickly due to time constraints. Key concept: Ito s lemma Stock Options: A contract giving
More informationTEST OF BOUNDED LOG-NORMAL PROCESS FOR OPTIONS PRICING
TEST OF BOUNDED LOG-NORMAL PROCESS FOR OPTIONS PRICING Semih Yön 1, Cafer Erhan Bozdağ 2 1,2 Department of Industrial Engineering, Istanbul Technical University, Macka Besiktas, 34367 Turkey Abstract.
More informationComputational Finance
Path Dependent Options Computational Finance School of Mathematics 2018 The Random Walk One of the main assumption of the Black-Scholes framework is that the underlying stock price follows a random walk
More informationOPTIMAL PORTFOLIO CONTROL WITH TRADING STRATEGIES OF FINITE
Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 005 Seville, Spain, December 1-15, 005 WeA11.6 OPTIMAL PORTFOLIO CONTROL WITH TRADING STRATEGIES OF
More informationOption Pricing Formula for Fuzzy Financial Market
Journal of Uncertain Systems Vol.2, No., pp.7-2, 28 Online at: www.jus.org.uk Option Pricing Formula for Fuzzy Financial Market Zhongfeng Qin, Xiang Li Department of Mathematical Sciences Tsinghua University,
More informationStochastic Processes and Stochastic Calculus - 9 Complete and Incomplete Market Models
Stochastic Processes and Stochastic Calculus - 9 Complete and Incomplete Market Models Eni Musta Università degli studi di Pisa San Miniato - 16 September 2016 Overview 1 Self-financing portfolio 2 Complete
More informationChapter 15: Jump Processes and Incomplete Markets. 1 Jumps as One Explanation of Incomplete Markets
Chapter 5: Jump Processes and Incomplete Markets Jumps as One Explanation of Incomplete Markets It is easy to argue that Brownian motion paths cannot model actual stock price movements properly in reality,
More informationStochastic Differential Equations in Finance and Monte Carlo Simulations
Stochastic Differential Equations in Finance and Department of Statistics and Modelling Science University of Strathclyde Glasgow, G1 1XH China 2009 Outline Stochastic Modelling in Asset Prices 1 Stochastic
More informationTHE USE OF NUMERAIRES IN MULTI-DIMENSIONAL BLACK- SCHOLES PARTIAL DIFFERENTIAL EQUATIONS. Hyong-chol O *, Yong-hwa Ro **, Ning Wan*** 1.
THE USE OF NUMERAIRES IN MULTI-DIMENSIONAL BLACK- SCHOLES PARTIAL DIFFERENTIAL EQUATIONS Hyong-chol O *, Yong-hwa Ro **, Ning Wan*** Abstract The change of numeraire gives very important computational
More informationLévy models in finance
Lévy models in finance Ernesto Mordecki Universidad de la República, Montevideo, Uruguay PASI - Guanajuato - June 2010 Summary General aim: describe jummp modelling in finace through some relevant issues.
More informationA No-Arbitrage Theorem for Uncertain Stock Model
Fuzzy Optim Decis Making manuscript No (will be inserted by the editor) A No-Arbitrage Theorem for Uncertain Stock Model Kai Yao Received: date / Accepted: date Abstract Stock model is used to describe
More informationRohini Kumar. Statistics and Applied Probability, UCSB (Joint work with J. Feng and J.-P. Fouque)
Small time asymptotics for fast mean-reverting stochastic volatility models Statistics and Applied Probability, UCSB (Joint work with J. Feng and J.-P. Fouque) March 11, 2011 Frontier Probability Days,
More informationCONVERGENCE OF NUMERICAL METHODS FOR VALUING PATH-DEPENDENT OPTIONS USING INTERPOLATION
CONVERGENCE OF NUMERICAL METHODS FOR VALUING PATH-DEPENDENT OPTIONS USING INTERPOLATION P.A. Forsyth Department of Computer Science University of Waterloo Waterloo, ON Canada N2L 3G1 E-mail: paforsyt@elora.math.uwaterloo.ca
More informationSTOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL
STOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL YOUNGGEUN YOO Abstract. Ito s lemma is often used in Ito calculus to find the differentials of a stochastic process that depends on time. This paper will introduce
More informationMAFS Computational Methods for Pricing Structured Products
MAFS550 - Computational Methods for Pricing Structured Products Solution to Homework Two Course instructor: Prof YK Kwok 1 Expand f(x 0 ) and f(x 0 x) at x 0 into Taylor series, where f(x 0 ) = f(x 0 )
More information- 1 - **** d(lns) = (µ (1/2)σ 2 )dt + σdw t
- 1 - **** These answers indicate the solutions to the 2014 exam questions. Obviously you should plot graphs where I have simply described the key features. It is important when plotting graphs to label
More informationThe End-of-the-Year Bonus: How to Optimally Reward a Trader?
The End-of-the-Year Bonus: How to Optimally Reward a Trader? Hyungsok Ahn Jeff Dewynne Philip Hua Antony Penaud Paul Wilmott February 14, 2 ABSTRACT Traders are compensated by bonuses, in addition to their
More informationMath 623 (IOE 623), Winter 2008: Final exam
Math 623 (IOE 623), Winter 2008: Final exam Name: Student ID: This is a closed book exam. You may bring up to ten one sided A4 pages of notes to the exam. You may also use a calculator but not its memory
More informationNumerical schemes for SDEs
Lecture 5 Numerical schemes for SDEs Lecture Notes by Jan Palczewski Computational Finance p. 1 A Stochastic Differential Equation (SDE) is an object of the following type dx t = a(t,x t )dt + b(t,x t
More informationContinuous-time Stochastic Control and Optimization with Financial Applications
Huyen Pham Continuous-time Stochastic Control and Optimization with Financial Applications 4y Springer Some elements of stochastic analysis 1 1.1 Stochastic processes 1 1.1.1 Filtration and processes 1
More informationSYSM 6304: Risk and Decision Analysis Lecture 6: Pricing and Hedging Financial Derivatives
SYSM 6304: Risk and Decision Analysis Lecture 6: Pricing and Hedging Financial Derivatives M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu October
More informationLocal vs Non-local Forward Equations for Option Pricing
Local vs Non-local Forward Equations for Option Pricing Rama Cont Yu Gu Abstract When the underlying asset is a continuous martingale, call option prices solve the Dupire equation, a forward parabolic
More informationMONTE CARLO METHODS FOR AMERICAN OPTIONS. Russel E. Caflisch Suneal Chaudhary
Proceedings of the 2004 Winter Simulation Conference R. G. Ingalls, M. D. Rossetti, J. S. Smith, and B. A. Peters, eds. MONTE CARLO METHODS FOR AMERICAN OPTIONS Russel E. Caflisch Suneal Chaudhary Mathematics
More informationShort-time-to-expiry expansion for a digital European put option under the CEV model. November 1, 2017
Short-time-to-expiry expansion for a digital European put option under the CEV model November 1, 2017 Abstract In this paper I present a short-time-to-expiry asymptotic series expansion for a digital European
More informationStochastic Volatility (Working Draft I)
Stochastic Volatility (Working Draft I) Paul J. Atzberger General comments or corrections should be sent to: paulatz@cims.nyu.edu 1 Introduction When using the Black-Scholes-Merton model to price derivative
More informationFast and accurate pricing of discretely monitored barrier options by numerical path integration
Comput Econ (27 3:143 151 DOI 1.17/s1614-7-991-5 Fast and accurate pricing of discretely monitored barrier options by numerical path integration Christian Skaug Arvid Naess Received: 23 December 25 / Accepted:
More informationMONTE CARLO BOUNDS FOR CALLABLE PRODUCTS WITH NON-ANALYTIC BREAK COSTS
MONTE CARLO BOUNDS FOR CALLABLE PRODUCTS WITH NON-ANALYTIC BREAK COSTS MARK S. JOSHI Abstract. The pricing of callable derivative products with complicated pay-offs is studied. A new method for finding
More informationEARLY EXERCISE OPTIONS: UPPER BOUNDS
EARLY EXERCISE OPTIONS: UPPER BOUNDS LEIF B.G. ANDERSEN AND MARK BROADIE Abstract. In this article, we discuss how to generate upper bounds for American or Bermudan securities by Monte Carlo methods. These
More informationPricing Barrier Options under Local Volatility
Abstract Pricing Barrier Options under Local Volatility Artur Sepp Mail: artursepp@hotmail.com, Web: www.hot.ee/seppar 16 November 2002 We study pricing under the local volatility. Our research is mainly
More informationNo-arbitrage theorem for multi-factor uncertain stock model with floating interest rate
Fuzzy Optim Decis Making 217 16:221 234 DOI 117/s17-16-9246-8 No-arbitrage theorem for multi-factor uncertain stock model with floating interest rate Xiaoyu Ji 1 Hua Ke 2 Published online: 17 May 216 Springer
More informationOption Pricing under Delay Geometric Brownian Motion with Regime Switching
Science Journal of Applied Mathematics and Statistics 2016; 4(6): 263-268 http://www.sciencepublishinggroup.com/j/sjams doi: 10.11648/j.sjams.20160406.13 ISSN: 2376-9491 (Print); ISSN: 2376-9513 (Online)
More informationSTOCHASTIC INTEGRALS
Stat 391/FinMath 346 Lecture 8 STOCHASTIC INTEGRALS X t = CONTINUOUS PROCESS θ t = PORTFOLIO: #X t HELD AT t { St : STOCK PRICE M t : MG W t : BROWNIAN MOTION DISCRETE TIME: = t < t 1
More informationHigh Frequency Trading in a Regime-switching Model. Yoontae Jeon
High Frequency Trading in a Regime-switching Model by Yoontae Jeon A thesis submitted in conformity with the requirements for the degree of Master of Science Graduate Department of Mathematics University
More information25857 Interest Rate Modelling
25857 UTS Business School University of Technology Sydney Chapter 20. Change of Numeraire May 15, 2014 1/36 Chapter 20. Change of Numeraire 1 The Radon-Nikodym Derivative 2 Option Pricing under Stochastic
More informationA SIMPLE DERIVATION OF AND IMPROVEMENTS TO JAMSHIDIAN S AND ROGERS UPPER BOUND METHODS FOR BERMUDAN OPTIONS
A SIMPLE DERIVATION OF AND IMPROVEMENTS TO JAMSHIDIAN S AND ROGERS UPPER BOUND METHODS FOR BERMUDAN OPTIONS MARK S. JOSHI Abstract. The additive method for upper bounds for Bermudan options is rephrased
More informationRisk Neutral Valuation
copyright 2012 Christian Fries 1 / 51 Risk Neutral Valuation Christian Fries Version 2.2 http://www.christian-fries.de/finmath April 19-20, 2012 copyright 2012 Christian Fries 2 / 51 Outline Notation Differential
More informationAnalytical formulas for local volatility model with stochastic. Mohammed Miri
Analytical formulas for local volatility model with stochastic rates Mohammed Miri Joint work with Eric Benhamou (Pricing Partners) and Emmanuel Gobet (Ecole Polytechnique Modeling and Managing Financial
More informationValuation of Asian Option. Qi An Jingjing Guo
Valuation of Asian Option Qi An Jingjing Guo CONTENT Asian option Pricing Monte Carlo simulation Conclusion ASIAN OPTION Definition of Asian option always emphasizes the gist that the payoff depends on
More informationSample Path Large Deviations and Optimal Importance Sampling for Stochastic Volatility Models
Sample Path Large Deviations and Optimal Importance Sampling for Stochastic Volatility Models Scott Robertson Carnegie Mellon University scottrob@andrew.cmu.edu http://www.math.cmu.edu/users/scottrob June
More informationAdvanced Topics in Derivative Pricing Models. Topic 4 - Variance products and volatility derivatives
Advanced Topics in Derivative Pricing Models Topic 4 - Variance products and volatility derivatives 4.1 Volatility trading and replication of variance swaps 4.2 Volatility swaps 4.3 Pricing of discrete
More informationMarket interest-rate models
Market interest-rate models Marco Marchioro www.marchioro.org November 24 th, 2012 Market interest-rate models 1 Lecture Summary No-arbitrage models Detailed example: Hull-White Monte Carlo simulations
More informationFinite Difference Approximation of Hedging Quantities in the Heston model
Finite Difference Approximation of Hedging Quantities in the Heston model Karel in t Hout Department of Mathematics and Computer cience, University of Antwerp, Middelheimlaan, 22 Antwerp, Belgium Abstract.
More informationLecture 17. The model is parametrized by the time period, δt, and three fixed constant parameters, v, σ and the riskless rate r.
Lecture 7 Overture to continuous models Before rigorously deriving the acclaimed Black-Scholes pricing formula for the value of a European option, we developed a substantial body of material, in continuous
More informationComputational Efficiency and Accuracy in the Valuation of Basket Options. Pengguo Wang 1
Computational Efficiency and Accuracy in the Valuation of Basket Options Pengguo Wang 1 Abstract The complexity involved in the pricing of American style basket options requires careful consideration of
More informationOption Pricing Models for European Options
Chapter 2 Option Pricing Models for European Options 2.1 Continuous-time Model: Black-Scholes Model 2.1.1 Black-Scholes Assumptions We list the assumptions that we make for most of this notes. 1. The underlying
More informationIlliquidity, Credit risk and Merton s model
Illiquidity, Credit risk and Merton s model (joint work with J. Dong and L. Korobenko) A. Deniz Sezer University of Calgary April 28, 2016 Merton s model of corporate debt A corporate bond is a contingent
More informationADAPTIVE PARTIAL DIFFERENTIAL EQUATION METHODS FOR OPTION PRICING
ADAPTIVE PARTIAL DIFFERENTIAL EQUATION METHODS FOR OPTION PRICING by Guanghuan Hou B.Sc., Zhejiang University, 2004 a project submitted in partial fulfillment of the requirements for the degree of Master
More informationThe Uncertain Volatility Model
The Uncertain Volatility Model Claude Martini, Antoine Jacquier July 14, 008 1 Black-Scholes and realised volatility What happens when a trader uses the Black-Scholes (BS in the sequel) formula to sell
More informationHedging under Arbitrage
Hedging under Arbitrage Johannes Ruf Columbia University, Department of Statistics Modeling and Managing Financial Risks January 12, 2011 Motivation Given: a frictionless market of stocks with continuous
More informationThe Black-Scholes Model
The Black-Scholes Model Liuren Wu Options Markets (Hull chapter: 12, 13, 14) Liuren Wu ( c ) The Black-Scholes Model colorhmoptions Markets 1 / 17 The Black-Scholes-Merton (BSM) model Black and Scholes
More information"Pricing Exotic Options using Strong Convergence Properties
Fourth Oxford / Princeton Workshop on Financial Mathematics "Pricing Exotic Options using Strong Convergence Properties Klaus E. Schmitz Abe schmitz@maths.ox.ac.uk www.maths.ox.ac.uk/~schmitz Prof. Mike
More informationFINITE DIFFERENCE METHODS
FINITE DIFFERENCE METHODS School of Mathematics 2013 OUTLINE Review 1 REVIEW Last time Today s Lecture OUTLINE Review 1 REVIEW Last time Today s Lecture 2 DISCRETISING THE PROBLEM Finite-difference approximations
More informationGreek parameters of nonlinear Black-Scholes equation
International Journal of Mathematics and Soft Computing Vol.5, No.2 (2015), 69-74. ISSN Print : 2249-3328 ISSN Online: 2319-5215 Greek parameters of nonlinear Black-Scholes equation Purity J. Kiptum 1,
More informationFE610 Stochastic Calculus for Financial Engineers. Stevens Institute of Technology
FE610 Stochastic Calculus for Financial Engineers Lecture 13. The Black-Scholes PDE Steve Yang Stevens Institute of Technology 04/25/2013 Outline 1 The Black-Scholes PDE 2 PDEs in Asset Pricing 3 Exotic
More informationSlides for DN2281, KTH 1
Slides for DN2281, KTH 1 January 28, 2014 1 Based on the lecture notes Stochastic and Partial Differential Equations with Adapted Numerics, by J. Carlsson, K.-S. Moon, A. Szepessy, R. Tempone, G. Zouraris.
More informationHedging Under Jump Diffusions with Transaction Costs. Peter Forsyth, Shannon Kennedy, Ken Vetzal University of Waterloo
Hedging Under Jump Diffusions with Transaction Costs Peter Forsyth, Shannon Kennedy, Ken Vetzal University of Waterloo Computational Finance Workshop, Shanghai, July 4, 2008 Overview Overview Single factor
More informationValuation of performance-dependent options in a Black- Scholes framework
Valuation of performance-dependent options in a Black- Scholes framework Thomas Gerstner, Markus Holtz Institut für Numerische Simulation, Universität Bonn, Germany Ralf Korn Fachbereich Mathematik, TU
More informationOption pricing in the stochastic volatility model of Barndorff-Nielsen and Shephard
Option pricing in the stochastic volatility model of Barndorff-Nielsen and Shephard Indifference pricing and the minimal entropy martingale measure Fred Espen Benth Centre of Mathematics for Applications
More informationNEWCASTLE UNIVERSITY SCHOOL OF MATHEMATICS, STATISTICS & PHYSICS SEMESTER 1 SPECIMEN 2 MAS3904. Stochastic Financial Modelling. Time allowed: 2 hours
NEWCASTLE UNIVERSITY SCHOOL OF MATHEMATICS, STATISTICS & PHYSICS SEMESTER 1 SPECIMEN 2 Stochastic Financial Modelling Time allowed: 2 hours Candidates should attempt all questions. Marks for each question
More informationUtility Indifference Pricing and Dynamic Programming Algorithm
Chapter 8 Utility Indifference ricing and Dynamic rogramming Algorithm In the Black-Scholes framework, we can perfectly replicate an option s payoff. However, it may not be true beyond the Black-Scholes
More informationPreface Objectives and Audience
Objectives and Audience In the past three decades, we have witnessed the phenomenal growth in the trading of financial derivatives and structured products in the financial markets around the globe and
More informationResearch Article Exponential Time Integration and Second-Order Difference Scheme for a Generalized Black-Scholes Equation
Applied Mathematics Volume 1, Article ID 796814, 1 pages doi:11155/1/796814 Research Article Exponential Time Integration and Second-Order Difference Scheme for a Generalized Black-Scholes Equation Zhongdi
More informationComputational Finance. Computational Finance p. 1
Computational Finance Computational Finance p. 1 Outline Binomial model: option pricing and optimal investment Monte Carlo techniques for pricing of options pricing of non-standard options improving accuracy
More informationContinuous Time Mean Variance Asset Allocation: A Time-consistent Strategy
Continuous Time Mean Variance Asset Allocation: A Time-consistent Strategy J. Wang, P.A. Forsyth October 24, 2009 Abstract We develop a numerical scheme for determining the optimal asset allocation strategy
More information